absorption kinetics on fractal surfaces

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Adsorption Kinetics on Fractal Surfaces

Massimiliano Giona and Manuela Giustiniani*,

Centro InteruniVersitario sui Sistemi Disordinati e sui Frattali nell Ingegneria Chimica, Dipartimento diIngegneria Chimica, UniVersita di Cagliari, Piazza dArmi, 09123 Cagliari, Italy, and Dipartimento diIngegneria Chimica, UniVersita di Roma La Sapienza, Via Eudossiana 18, 00184 Roma, Italy

ReceiVed: May 23, 1996X

A model is proposed for the description of diffusion-controlled adsorption kinetics on fractal surfaces. Thismodel is based on a constitutive equation between the mass flux and the concentration gradient of the adsorbingspecies expressed in terms of a Riemann-Liouville (fractional) operator of noninteger order . The order depends on the fractal dimension df of the adsorbent surface, ) df - dT, dT being its topological dimension.The model is compared with Monte Carlo simulations and with the approach proposed by Seri-Levy andAvnir and displays a good level of agreement with Monte Carlo data over all time scales.

1. Introduction

In a recent paper, Seri-Levy and Avnir1 proposed a modelfor adsorption kinetics in diffusion-limited conditions. This

model is based on an extension of Delahays analysis,2-4

whichis valid for flat surfaces, so as to include the effects of fractality.In the article by Seri-Levy and Avnir, the resulting expressionfor the fractional coverage (t)/e vs time t is a modificationof the equation derived by Delahay and Trachtenberg2 in orderto include the anomalous scaling of the adsorbed volume withthe fractal dimension. The model proposed in ref 1 constitutesa significant initial contribution toward the development of anapproximate macroscopic model for the kinetics of adsorptionon fractal interfaces. The derivation of approximate mean-fieldmodels taking advantage of phenomenological and experimentalstudies and of the results of scaling theories for fractal structureshas great importance, especially as regards practical engineeringproblems, and contributes toward a better understanding of the

physics of fractals.The vast literature on mass-transfer kinetics across fractalinterfaces and membranes is primarily oriented toward thederivation of anomalous scaling laws and interpretation of thestructural properties of the interfaces (see for a review refs 5-7).

Most of the literature on this topic is based on the analogybetween membrane and interfacial kinetics and the electricresponse of rough electrodes, for which an anomalous imped-ance scaling with frequency (referred to as constant phaseangle behavior, or CPA), Z() ) R0 + R(i)-, 0 < < 1, i

) -1, has been observed.8-12 The exponent depends onthe roughness of the electrode and ultimately on the fractaldimension df. See ref 6 for a brief analysis of the differentmodels proposed for the relationship between and df.

An initial attempt to derive a macroscopic theory fromphenomenological scaling relations on the properties of roughelectrodes was made by Le Mehaute,13 who introduced aconvolutional relation expressed in terms of Riemann-Liouvilleintegrals (fractional operators)14,15 between fluxes and drivingforces (concentration gradients). Macroscopic models fordiffusion on fractal media involving fractional operators havebeen proposed by Giona and Roman16,17 and by Nigmatullin18,19

and are widely applied in modeling the rheological propertiesof viscoelastic materials20 and the relaxation properties ofdielectrics.21-23

This article develops a macroscopic (approximate) model fordiffusion-limited adsorption kinetics on fractal structures basedon a constitutive equation between fluxes and concentrationgradients expressed in terms of Riemann-Liouville operators.The primary concept involved in the application of Riemann-Liouville operators in this case is the fact that the presence ofsurface roughness locally modifies the diffusion properties ofthe adsorbing species. Ficks law cannot therefore be directlyapplied in the neighborhood of the interface, and correlationproperties arise as a consequence of diffusion and trapping(adsorption) in particle motion. These correlations can bemacroscopically modeled as memory effects in particle motionand can therefore be expressed by means of a constitutiveequation of convolutional nature.

The article is organized as follows. First we present anextensive Monte Carlo simulation quite close to saturation (-(t)/e = 0.8) on a self-similar fractal structure, following thetechnique discussed in ref 1. The approach adopted by Seri-Levy and Avnir is reviewed. We then propose a macroscopicmodel of adsorption kinetics by introducing a fractionalconstitutive equation between fluxes and concentration gradients.Two cases are considered: the idealized case of an infinitelyextended fractal layer and the more realistic case of a fractallayer of finite width. The exact meaning of these two cases isextensively discussed below. The theory presented is subjectedto critical analysis and compared both with Monte Carlo dataand with the equation proposed by Seri-Levy and Avnir.

2. Monte Carlo Simulations

Two-dimensional simulations of diffusion-controlled adsorp-tion kinetics on a fractal interface were performed by means of

the Monte Carlo method on a square lattice according to thefollowing rules.1 Nmol adsorptive molecules are randomlydistributed in a periodic reservoir. The reservoir size is suchthat bulk conditions can be approached. The adsorbent surfaceis placed in the middle of the reservoir. At each step (a) amolecule is chosen at random; (b) if it is adjacent to anunoccupied surface site, it is adsorbed with probability pa; (c)if the molecule is already adsorbed, it can desorbe withprobability pd. Steps a-c are repeated Nmol times for eachphysical time. Adparticles are noninteracting unless throughexcluded-volume effects.

To keep the bulk concentration constant, the Ne particlesadsorbed at each time are counted and Ne unadsorbed moleculesare added and randomly distributed in the bulk region.

Universita di Cagliari. Universita di Roma La Sapienza.X Abstract published in AdVance ACS Abstracts, September 1, 1996.

16690 J. Phys. Chem. 1996, 100, 16690-16699

S0022-3654(96)01518-3 CCC: $12.00 1996 American Chemical Society


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the fractal interface with fractal dimension df ) 1.5, obtainedunder the simulation conditions Nmol ) 30 000, Nsites ) 819 200,reservoir size 51 200 1200 lattice units, corresponding to abulk concentration ofc0 ) 4.88 10-4, K) 20, and diffusioncoefficient D ) 1. The model, eq 7, with D ) 1 and k )

pa/pd ) 20, is represented by the solid line. Identity between kand Kwas analyzed by Seri-Levy and Avnir. A good level ofagreement is observed up to coverages of about (t)/e ) 0.45.

This result is qualitatively the same as that reported by Seri-Levy and Avnir up to coverages of (t)/

e) 0.45. Direct

quantitative comparison is not possible because the normalizingvalue e is different. This was calculated by Seri-Levy andAvnir by means of a Monte Carlo simulation of the equilibriumconditions at a bulk concentration c0, which is higher than Kc0.

We assume that the adsorption isotherm, eq 5, is independentof the surface roughness, i.e. that the Henry constant K in eq 5is independent of the surface fractal dimension. This is in linewith experimental observations and with several theoreticalworks24,25 which point out that the influence of geometric (andenergetic) heterogeneities is negligible at low coverages. Themaximum coverage is therefore the same as in the flat case,i.e. e ) Kc0, and at low coverages the surface roughness isassumed to influence only the diffusional phenomena adjacent

to it.The good level of agreement between simulation data andmodel prediction shown in Figure 2 is no longer observed atlarge time scales, as is evident in Figure 3, which shows thesame data as in Figure 2 but in the broader range t [0,200 000]. From Figure 3, it is evident that agreement betweenmodel and simulation data is restricted to short timee scalesand that the model predicts slower adsorption kinetics than thoseobserved from Monte Carlo simulations.

4. Fractional Diffusion Equation

The theory for adsorption kinetics on fractal interfaces basedon Riemann-Liouville (fractional) constitutive equations isdeveloped over this section and the next. To develop the theory

without discontinuities in the analysis, comparison with MonteCarlo experiments is postponed to section 6.

Let us start with a phenomenological consideration. In theirstudies on the electric response of fractal electrodes, Pajkossyand Nyikos26,27 found that the flux of the transported entity Jon a fractal interface behaves as follows:

where the exponent is given by

Equations 8 and 9 correspond to a situation in which theconcentration of the transported entity can be assumed to beequal to zero at the interface and the bulk phase concentrationis initially constant. In this case, the classical diffusion equationpredicts an exponent ) 1/2, which is recovered for df ) dT,i.e. in the limit of a flat interface (surface).

It is evident that eqs 8 and 9 cannot be predicted by meansof the classical Ficks law, i.e. in terms of a flux that isproportional to the concentration gradient through a constant.A generalized approach to the problem of transport through

fractal interfaces should therefore be attempted in terms oftemporal correlations in the constitutive equation between theflux and the concentration gradient of the transported species.

Memory effects in diffusion through non-homogeneous mediahave been theoretically studied by Goddard,28 who shows thatthis phenomenon can be represented by introducing memorykernels and convolution integrals in time to describe the historyeffects arising from microscopic diffusional relaxation. Frac-tional diffusion equations have been applied by Giona andRoman16,17 and by Nigmatullin18,19 to model diffusion in fractalmedia macroscopically and find application in the study of therelaxation of dielectric materials,21-23 in the rheology ofviscoelasticity,29-33 and in general in connection with correlatedrandom walks.34-37

In terms of the macroscopic approximate balance equations,the constitutive equation for the flux-concentration gradientdependence underlying memory effects can be expressed bymeans of a Riemann-Liouville integral operator with acharacteristic noninteger order depending on the adsorbentfractal dimension.

This model applies to the typical phenomenology occurringin electrochemistry and adsorption, i.e. diffusion in a bulk phasewith adsorption on a surface, which was first investigated byDelahay et al.2 and further studied for fractal structures by Seri-Levy and Avnir.1

To describe a diffusion-controlled phenomenon in the pres-ence of fractal interfaces, a generalized diffusion equation isproposed of the following form:

where x is the vector of the spatial coordinates, t is the timevariable, c(x, t) is the concentration of the adsorbing species inthe bulk phase, and the convolutional operator * is applied to adiffusiVity memory kernel D F(t). It should be pointed out thatD F is not, strictly speaking, a diffusivity, but rather a diffusivity-dependent parameter.

Equation 10 is equivalent to the assumption of the generalizedflux-concentration constitutive equation

The functional form of the diffusivity kernel D F(t) is derivedbelow and is physically justified.

In the following, we refer always to a one-dimensionaltransport model; that is, eqs 10 and 11 simplify as follows:

or equivalently

Figure 3. The same as Figure 2 observed at longer time scales.

J t- (8)

)df - dT + 1

2(9)

c(x,t)

t) D F(t) *

2c(x, t) (10)

J) -D F(t) * c(x, t) (11)

c(x, t)

t) D F(t) *

2c(x, t)

x2

(12)

J(x, t) ) -D F *c(x, t)

x(13)

16692 J. Phys. Chem., Vol. 100, No. 41, 1996 Giona and Giustiniani


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where x is the coordinate in the direction of transport (roughlyspeaking, macroscopically orthogonal to the interface).

It should be observed that in the flat case flux and concentra-tion are rigorously calculated at x ) 0, the adsorbent surfacebeing really identified at x ) 0. On the contrary, for a fractaladsorbent surface, a virtual line at x ) 0, acting as a planarprojection of the rough interface, must be considered in orderto impose the boundary conditions. This is made necessary bythe fact that a fractal curve is nondifferentiable almost every-where, and the direction orthogonal to the surface is therefore

not defined at each point. This schematization represents theintrinsic approximation of any macroscopic model (balanceequation) involving fractal (nondifferentiable) structures.

Before applying eq 12 to diffusion-controlled adsorptionkinetics, we shall derive the functional form of D F(t) by makinguse of the results expressed by eqs 8 and 9. The boundary andinitial conditions associated with the model of Pajkossy andNyikos26 are c(0, t) ) 0, c(, t) ) c0 ) constant for t > 0 andc(x, 0) ) c0 for x > 0.

With these conditions, eq 12 can be solved in the Laplacedomain to yield

where the tilde symbol identifies the Laplace transforms of thefunctions of time c(x, t), D F(t), and s is the Laplace variable.

The flux at the interface is therefore given in the Laplacedomain by

where L [ ] indicates the Laplace transform. Equation 15 canbe compared with the anomalous diffusion scaling law obtainedfor fractal interfaces, eq 8, in order to derive a functional formfor the diffusivity kernel D F(s) as a function of s. In theLaplace domain, eq 8 becomes

Cbeing a generic constant. To satisfy eqs 8 and 9, the Laplacetransform of the kernel D F(s) must therefore satisfy theequation

where the exponent is given by

The constant D F0 has the dimensions [m] 2[s]df-dT-1 and is,

strictly speaking, a diffusivity only if df ) dT, i.e. in thenonfractal case. Since eq 17 is derived from a scaling relation,eq 8, its validity is of course restricted to a given range of svalues. We shall return to this topic below.

Equation 17 can be regarded as an intrinsic property of themodel, independently of the particular equilibrium conditionassumed at the interface, thus making it possible to describethe general power law behavior observed for diffusion throughfractal interfaces. It should be observed that eq 17 has asingularity, corresponding to the condition dfg dT, i.e. g 0.

The inverse of the Laplace transform of the flux can beobtained in terms of fractional derivatives.15 By substitutingeq 17 into the expression for the Laplace transform of the flux,we obtain

The inverse Laplace transform of eq 19 can be expressed interms of Riemann-Liouville integral operator and is given by(see Appendix I)

where (1 - ) is the gamma function of argument 1 - .Equation 20 represents the explicit form of the constitutiveequation assumed for the dependence of the flux on theconcentration gradient, i.e.

Having derived the functional form of the diffusivity kernel,we shall go on to analyze the kinetics of adsorption in thepresence of diffusional control, i.e. in the Delahay hypothesisfor a fractal interface.

As in the flat case discussed in section 3, two phasesare assumed: a fluid phase with concentration c(x, t) and anadsorbed phase with a surface concentration (t), defined byeq 1. The boundary conditions are the same as in the flat case,bearing in mind that in eq 1 the expression of the interfacialflux is given by eq 20 and that the equation describing diffusionis given by eq 12.

By solving the balance equations with the given initial andboundary conditions, one obtains for the Laplace transforms(s) the expression

where e, as previously defined, is e ) Kc0. Substitution ofeq 17 into eq 22 leads to the model solution in the Laplace

domain:

The limiting case of a flat adsorbent surface is recovered for df) dT, i.e. ) 0, and D F0 ) D .

Equation 23 can be analytically inverted by means of thecomplex inversion formula, as shown in Appendix II, thusobtaining

where the notation has been simplified by replacing ) (1 -)/2, and y is a real variable.

It can be observed that the limiting flat case is satisfied for ) 1/2 (i.e. ) 0), and eq 6 is recovered for D F0 ) D .Moreover, eq 24 satisfies the conditions (0) ) 0 and () )e ) Kc0.

Here we anticipate an important point discussed extensivelyin section 6.

The results obtained from the model expressed by eq 17, orequivalently by eq 24, are not as satisfactory at long time scales

c(x, s) )c

0

s [1 - exp[-(s

D F(s))1/2

x]] (14)

L [J]x)0 ) D F(s)(cx

)x)0

) c0(D F(s)

s)

1/2

(15)

L [J] ) Cs-1

(16)

D F(s) ) DF0s2-1

) DF0s (17)

) df - dT (18)

L [J]x)0 ) -DF0s(c

x)x)0 (19)

J|x)0 ) -DF0

(1 - )

t0

t(t- )-(c(x, )

x )x)0 d (20)

J(x, t) ) -DF0

(1 - )

t0

t(t- )-

c(x, )

xd (21)

(s)

e

)D F(s)/K

s(s + D F(s)/K)(22)

(s)

e

)D F0/K

s(s(1-)/2

+ D F0/K)(23)

(t)

e

)

1 - D F0

K0 exp[-ty1/] sin()

(y cos() + D F0/K)2

+ y2

sin2()

dy (24)

Adsorption Kinetics on Fractal Surfaces J. Phys. Chem., Vol. 100, No. 41, 1996 16693


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as at short time scales. A good level of agreement betweenmodel and simulation data is observed only at short time scales.

To some extent, this can be accounted for by pointing outthat the effect of adsorbent roughness is expected to exercise astrong influence upon short-term behavior.

This is a consequence of the fact that the spatial region inwhich diffusive motion is dominated by anomalies is restrictedto the neighborhood of the fractal interface. At a distance fromthe interface, diffusion is still regular since the influence offractality is negligible for x . 0.

This effect can be accounted for either by describing it interms of two temporal length scales or by considering the factthat the spatial region over which diffusion is anomalous has afinite width. The former approach is described in this section,and the latter, which leads to the concept of the fractal layer, inthe next.

Let us consider the first model formulation. Two differenttemporal behaviors can be identified and distinguished. At shorttime scales, the fractality of the adsorbent surface stronglyaffects the resulting adsorption kinetics, whereas interfaceroughness is be a negligible factor at long time scales.

Starting from this observation, the model, eq 11, can bemodified in order to describe the existence of these twocharacteristic temporal regions. The constitutive equation can

be regarded as a superposition of the two characteristicbehaviors:

where the first term on the right-hand side is the Fickiancontribution significant for large time scales (i.e. far from theinterface) and the second one is the anomalous constitutiveequation previously defined.

The governing equation is obtained by substituting thisexpression into the balance equation c/t) -J/x (or in vectorform c/t ) -J).

Equation 25 corresponds in the Laplace domain to the

assumption of an equiV

alent diffusiV

ity kernel of the form

where eq 17 still gives the diffusivity kernel D F(s). It shouldbe observed that eq 26 is still consistent with eq 8 since thelatter is strictly a scaling relation.

The resulting solution in the Laplace domain can thereforebe obtained in the form of eq 22, where the equivalent diffusivityD eq(s) rather than D F(s) appears, i.e.

Together with eq 26, eq 27 can be analytically inverted back tothe time domain. The details can be found in Appendix II.

In the case of the Koch curve with fractal dimension df )1.5 considered in the simulations, the solution attains thefollowing form in the time domain:

where A ) D F0/K2, B ) D 0/K2, z is a real variable, and tan()) Az/B.

The parameters D F0 and D 0 must of course depend on thebulk diffusion D in the following way:

q0 being a dimensionless constant and q1 a constant withdimension [m]2(1-).

As will be discussed in the following sections, very goodresults are obtained by correlating eq 28 with the simulationdata.

It should be pointed out that the proposed model does nottake into account the influence of the fractal nature of theadsorbent structure on the local adsorption equilibrium. Thesame linear adsorption isotherm has therefore been applied asin the flat case. This assumption is theoretically justified onthe grounds that the concentrations in the bulk and adsorbedphases are very low. In these conditions, the effect of surfaceroughness is expected to be negligible, as suggested in otherworks and as discussed in section 2.

5. Fractal Layer

The macroscopic adsorption kinetic model expressed by eqs27 and 28 represents the role of the adsorbent roughness in termsof a memory effect by means of the characteristic fractal

dimension dependent exponent . This constitutes a globalmodeling of different effects due to roughness. In general,roughness gives a higher adsorbent site number than the planarprojection of the interface, different site accessibility, and theexistence of a nonzero thickness influencing diffusion close tothe interface.

The model of eq 27 neglects the fact that a fractal interfacehas a finite thickness. A step toward the description of thispoint is given by an extended version of the proposed modelcharacterized by three phases: (i) a bulk region, far from thefractal interface, in which the diffusional phenomena are regular,i.e. not influenced by the fractality of the solid surface, andFicks law applies; (ii) a fractal region (as referred to the fractallayer), in which anomalous diffusion occurs, described by means

of model eq 10; (iii) an adsorbed surface phase. A finitethickness of the fractal interface LF is thus introduced, delimitingthe region in which anomalous diffusion occurs.

The resulting balance equations should therefore read asfollows.

1. In the bulk phase (z [0, ))

where D is the diffusion coefficient.2. In the fractal layer with thickness LF, q(x, t) being the

concentration in this phase,

where x [0, LF]. The spatial coordinates x and z are of courserelated by z ) x - LF.

3. In the adsorbed phase, characterized by a surfaceconcentration,

The boundary and initial conditions are, as in the case of flatadsorbents, given by c(, t) ) c0, (t) ) Kq(0, t) for t> 0, c(z,0) ) c0 for z (0, ), q(x, 0) ) c0 for x (0, LF). The

J(x, t) ) -D 0c(x, t)

x- D F(t) *

c(x,t)

x(25)

D eq(s) ) D 0 + D F(s) ) D 0 + D F0s (26)

(s)

e

)D eq(s)/K

s(s + D eq(s)/K)(27)

(t)

e

) 1 +2

0

exp(-z2t)[ Az

2

(B + z2)

2+A

2z

2-

(B2

+A2z

2)

1/4[(B + z

2) cos(/2) + Az sin(/2)]

(B + z2)

2+ A

2z

2 ] dz (28)

D 0 ) q0D D F0 ) q1D (29)

c

t) D

2c

z2

(30)

q

t) D F *

2q

x2

(31)

(t) )0tD F * (

q

x)x)0 d (32)



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continuity between the fractal layer and the bulk phase impliesthat

It is evident that the model discussed in the previous sectioncorresponds to the particular case for LF f , and the planaradsorbent surface is recovered if LF ) 0.

The solution of the model in the Laplace domain is then

obtained:

where M and J are defined as

It should be observed that the Laplace transform of the solutionof the model, eq 34, satisfies the particular cases for LF ) 0and LF f .

In the case LF f , eq 23 is recovered from eq 34; in thecase LF ) 0, eq 34 yields

which is the Laplace transform of the solution in the flatinterface case, eq 6. Moreover, eq 34 satisfies the condition(0) ) 0.

The description of simulation data by means of eq 34 provesquite satisfactory, as will be shown in the next section.

6. Results

A convenient way to compare the proposed models withMonte Carlo results is to use eqs 22, 27, and 34 directly in theLaplace domain. We therefore calculated the Laplace transformof an interpolating function of the adsorption curves vs time.The interpolating function for the Monte Carlo data obtainedon the Koch fractal interface (df ) 1.5) was chosen in the form

f(t) ) 1 - exp[-at0.25 - bt0.5]. This interpolation satisfies thecharacteristic short-time behavior for the interface with fractaldimension df ) 1.5, i.e. J df(t)/dt t0.25.

From eq 22, an effective diffusivity D eff(s) can be definedas

This representation allows direct comparison of the modelsdiscussed in the previous sections and Monte Carlo simulations.

Figure 4 shows the simulation data (]) already reported inFigures 2 and 3 (simulation conditions: Nmol ) 30 000, Nsites) 819 200, reservoir size 51 200 1200 lattice units, c0 )4.88 10-4, K ) 20) together with the model, eq 23, withD F0 ) 0.71 (solid line a). In this case, D eff(s) ) D F(s). Thislevel of agreement is fairly poor at long time scales, i.e. at s f0, and prediction in time domain is comparable with the resultsobtained by means of the Seri-Levy/Avnir model.

Figure 4 also shows the improvement in the description of

the same simulation data achieved by means of the equivalent-diffusivity model eq 26 (solid line b) and the fractal-layer model(solid line c), where the effective diffusivity defined in eq 37 isplotted against s. The values of the parameter obtained fromcurve fitting are respectively D 0 ) 3.5 10-3 and D F0 ) 0.71for the equivalent-diffusivity model and DF0 ) 0.73 and LF )11 for the fractal-layer model. Since D ) 1, eq 29 yields thevalues q0 ) 3.5 10-3 and q1 ) 0.71 for the parameters qi.

The level of agreement is satisfactory for both the models,as is evident in Figure 5a, where the inverse of the Laplacetransform of the model, eq 28 (solid line), is shown togetherwith simulation data (dots) up to 600 000 time steps. In Figure5a, the normalized variables ()/e and ) tD /L2, with L )1 lu are reported respectively on the y and x axes. The average

c(0, t) ) q(LF, t) D (c

z)z)0 ) D F * (q

x)x)LF (33)

(s)

e

)1

s+

K

s

(J+ 1) - (J- 1) exp[-2LF/M]

exp[-2LF/M](J- 1)(M- K) - (J+ 1)(M+ K)(34)

M)D F(s)

sJ)

D F(s)

D(35)

(s)

e

)D /K

s(s + D /K)(36)

D eff(s) )s32(s)K2

(e - s(s))2

(37)

Figure 4. D eff(s) vs s. The proposed models are compared withMonte Carlo simulations (]). Solid lines represent (a) eq 23 with D F0) 0.71, (b) eq 26 with D 0 ) 3.5 10-3 and D F0 ) 0.71, and (c) thefractal-layer model, eq 34, with D F0 ) 0.73, LF ) 11. The simulationparameters are Nmol ) 30 000, Nsites ) 819 200, reservoir size 51 200 1200 lu, c0 ) 4.88 10-4, K ) 20, and D ) 1.

Figure 5. (t)/e vs . Comparison of Monte Carlo simulations (dots)and the model, eq 28 (solid line), with D 0 ) 3.5 10-3 and D F0 )0.71. The simulation parameters are Nmol ) 30 000, Nsites ) 819 200,reservoir size 51 200 1200 lu, c0 ) 4.88 10-4, K ) 20, and D )1. (a, top) Up to 600 000 time steps; (b, bottom) up to 50 000 timesteps.



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error, calculated as 1/Ni)1N|ytheor - ysim|/ytheor, is 1.2% over

751 points.In Figure 5b, the good level of agreement between the model

and Monte Carlo simulations is shown in the restricted range [0, 50 000].

Analogous results were obtained for the same fractal interfaceat the third iteration. In this case, the thickness of the fractalregion is, of course, smaller than at the fourth iteration (42 asagainst 170 lattice units). The simulation conditions are Nmol) 20 000, reservoir size 51 200 1200, Nsites ) 409 600, c0 )3.25 10-4, K ) 20, and D ) 1.

Figure 6 shows the comparison of the fractal-layer (solid linea) and equivalent-diffusivity (solid line b) models with simula-tions in the Laplace domain in terms of effective diffusivity,eq 37.

For the equivalent-diffusivity model, the values of theparameters obtained are D F0 ) 0.9 and D 0 ) 0.013. It is

interesting to observe that D 0 is strictly related to the thicknessof the fractal interface. As the iteration in the generation ofthe structure increases, and the thickness of the interfaceincreases correspondingly, the parameter D 0 becomes smaller.This is in agreement with the observation that the case LF f of the fractal-layer model, eq 34, corresponds directly to theparticular case D 0 ) 0 in eq 26, i.e. to eq 23.

For the fractal-layer model, eq 34 was used to find the bestfit for LF ) 7, D0 ) 0.95. From Figure 6, it is evident thatboth models provide a good description of the Monte Carlosimulation.

Figure 7 shows the inverse Laplace transform of theequivalent-diffusivity model, eq 28, with D F0 ) 0.9 and D 0 )0.013 (solid line), together with simulation data (dots). The

dimensionless variables ()/e and , where ) D t/L2, L ) 1lu, are reported as the x and y axes. The average error is about2% over 2000 points. The higher fluctuations in Monte Carlodata are due to the smaller number of particles Nmol in thesimulation.

The model analysis can be completed by examining thebehavior for different Henrys adsorption equilibrium constantsK and for different diffusivities in the bulk fluid phase D .

Figure 8a,b presents a comparison in the time domain ofMonte Carlo data with the equivalent-diffusivity model, eq 28,for different values of the Henrys constant K. In Figure 8a,dots are used to represent the Monte Carlo results obtained withthe following simulation conditions: Nmol ) 100 000, reservoirsize 51 200 1200 (resulting concentration c0 ) 1.63 10-3),

D ) 1, and K ) 2 on the Koch fractal interface with df ) 1.5(fourth iteration). The model predictions (solid line) correspond

to the values of the parameters previously obtained, i.e. D 0 )3.5 10-3 and D F0 ) 0.71, K ) 2, and can be consideredfairly satisfactory.

Figure 8b refers to the same simulation conditions as in Figure8a with a different value of the Henrys constant, K ) 10. Thelevel of agreement between model (solid line is eq 28) andsimulations (dots) is very good.

In Figure 8a,b the characteristic model parameters appearingin eq 28 are the same as in Figure 5. This confirms that theassumed model of the equivalent diffusivity, eq 28, simplydescribes anomalous diffusion due to the fractal nature of theadsorbent surface, the parameters D 0 and D F0 being independentof the equilibrium Henrys constant.

The same conclusion can be drawn by analyzing the adsorp-

Figure 6. D eff(s) vs s. The proposed models are compared withMonte Carlo simulations (]). The simulation conditions are Nmol )20 000, Nsites ) 409 600, reservoir size 51 200 1200 lu, c0 ) 3.25 10-4, K ) 20, and D ) 1. The adsorbent surface is the Koch fractalcurve (df ) 1.5) at the third iteration. Solid lines represent (a) the fractallayer model and (b) the equivalent diffusivity model, eq 26. Fittingparameters: (a) D F0 ) 0.95 and LF ) 7; (b) D F0 ) 0.9 and D 0 )0.013.

Figure 7. ()/e vs . Comparison of Monte Carlo simulations (dots)and the model, eq 28 (solid line), with D 0 ) 0.013 and D F0 ) 0.9.The simulation parameters are Nmol ) 20 000, Nsites ) 409 600, reservoirsize 51 200 1200 lu, c0 ) 3.25 10-4, K ) 20, and D ) 1. Theadsorbent surface is the Koch fractal curve at the third iteration.

Figure 8. ()/e vs at different values of Henrys constant.Simulation data: Nmol ) 100 000, Nsites ) 819 200, reservoir size 51 200 1200 lu, c0 ) 1.63 10-3, D ) 1, and (a, top) K ) 2, (b, bottom)K ) 10. The solid line is the prediction obtained by the model, eq 28,with the same parameters as Figure 5, D 0 ) 3.5 10-3 and D F0 )0.71.



8/10

tion curves obtained at different values of diffusivity in the bulkphase. Figure 9 shows simulation (Nmol ) 100 000, reservoirsize 51 200 1200 lu) and model predictions for D ) 10 andK ) 10. The model is fully predictive since the parametersD 0 and D F0 are obtained by applying eq 29 with the values ofq

0and q

1derived from Figure 4, i.e. D

0) 3.5 10-2 and D

F0) 2.25.

As in the case of a flat adsorbent surface, the adsorption curvevs the dimensionless time ) tD /L2, with L ) 1 lu, isindependent of the bulk diffusion coefficient D . This can bedirectly obtained from the functional form of the model, eq 28,which depends on D only through the dimensionless group ) tD /L2. This is also confirmed by the simulations, byobserving that the adsorption curves obtained for D ) 1 andD ) 10 coincide for the same values of the other simulationparameters.

To sum up, both the equivalent-diffusivity model and thefractal-layer model furnish satisfactory predictions of kineticdata. In the case of the equivalent-diffusivity model, a complete

analysis of the functional dependence of the model parameterson K and D has been carried out and is summarized in eq 29.

The only topic requiring further investigation is the behaviorof the fractal-layer width LF with respect to the structural(geometric width of the interface) and dynamic parameters, i.e.D . Unfortunately, such analysis requires extremely large lattice(reservoir) size and is beyond our present capacity. Thisimportant topic will hopefully be analyzed elsewhere.

7. Sorption Kinetics: Scaling Observations

This section focuses on an interesting analogy between thesorption behavior of volume fractals and the adsorption proper-ties of fractal interfaces.

Giona et al.38 recently analyzed the sorption properties ofvolume fractals by applying exact Green function renormaliza-tion. More specifically, a fractal structure, in which theconcentration of the diffusing species is initially zero, is placedin contact with a reservoir (of infinity capacity) at constantconcentration c0. This situation corresponds to sorption experi-ments customarily performed in membrane science, in the studyof microporous materials,43 and in the study of the controlledrelease of swollen polymeric matrices.42 Solute motion isassumed to be purely diffusive within the fractal structure.

The quantity of interest as providing a macroscopic descrip-tion of sorption kinetics is the ratio between the quantity ofsolute entering the structure up to time t and the total quantityentering the structure asymptotically (i.e. for tf ).

In the case of bulk fractals (e.g. the Sierpinski gasket), theresults obtained by Giona et al. can be summarized by thescaling relation

where dw is the walk dimension (dw ) 2 for Euclidean media,dw > 2 for fractals), df the fractal dimension of the structure,and dfb the dimension of the transfer surface exposed to the

external reservoir concentration c0. In particular, if the exchangearea reduces to a single point, dfb ) 0, and eq 38 becomes Mt/

M tds/2, where ds is the spectral (fracton) dimension, since2/dw ) ds/df. In the case dfb ) df - 1, Mt/M t1/dw.

It is interesting to observe that eq 38, which was obtainedby applying exact renormalization to bulk fractals, also describesthe sorption properties of fractal interfaces. Indeed, from eqs8 and 9 one obtains

where dfs is used to indicate the fractal dimension of the

interface in order to avoid confusion with the fractal dimensiondf of the bulk entering into eq 38.

Let us compare eq 39 with eq 38. In the case of adsorptionon a fractal surface, diffusion takes place on a regular Euclideanmedium with fractal dimension df ) dT + 1 and with walkdimension dw ) 2. In this case, the transfer area dimension dfbcoincides with the fractal dimension of the interface df

s. Thisindicates that eq 38 is a unifying relation that describes thesorption properties of both bulk fractals and fractal interfaces.

The result expressed by eq 38 shows that the quantity Mt/M, which is customarily measured in sorption experiments,depends strongly on the dimension of the exposed surface dfb.Different scaling behaviors may therefore be expected as aconsequence of different distributions of adsorbing centers, e.g.in the case of a multifractal distribution, as discussed byGutfraind et al.41 in connection with catalytic properties.

8. Concluding Remarks

This article proposes a macroscopic theory for the kineticsof adsorption on fractal structures. Although the analysis isfocused exclusively on the diffusion-controlled regime, it shouldbe borne in mind that the theory applies to a broader phenom-enology since different regimes are mathematically describedby the boundary condition at x ) 0.

Indeed, in the case of linear transfer kinetics at the interface,the boundary conditions become

Figure 9. (t)/e vs t at bulk diffusivity D ) 10. The Monte Carlosimulation (dots) is compared with the model prediction, eq 28 (solidline). The simulation conditions are Nmol ) 100 000, c0 ) 1.63 10-4,Nsites ) 819 200, and K ) 10, and the model parameters q0 ) 3.5 10-3 and q1 ) 0.71.

Figure 10. Domain of the complex plane used to obtain the Laplaceinverse, eq 23 and eq 27.

Mt/M t(df-dfb)/dw (38)

Mt/M 0tJ() d t(1+dT-df

s)/2(39)



9/10

where h is the mass-transfer coefficient and eq 12 can be solvedwith these conditions. The diffusion-controlled regime can beobtained from eq 40 for h f .

By analogy, the representation of the anomalies in diffusion/adsorption kinetics of fractal interfaces by means of fractionaloperators finds confirmation in the exact results obtained in thecase of diffusion on fractal media by applying Green function

renormalization.38 In the case of sorption kinetics on fractals, apower-law behavior in s was observed for the Laplace trans-forms of the Green functions.

Both the phenomenologies, i.e. (1) adsorption across a fractalinterface and (2) sorption kinetics of bulk fractals, can be unifiedby means of a single scaling relation, eq 38, in which theexponent depends on the dimension of the exchange area dfb,on the fractal dimension of the bulk structure df, and on thewalk dimension dw.

To achieve a complete understanding of diffusion acrossfractal interfaces and to overcome the intrinsic limitations ofMonte Carlo simulations, it would of course be highly advisableto extend to this case the Green-function renormalizationsuccessfully applied in the case of diffusion and adsorption onfractal media.

This article introduces the concept of the fractal layer, whichcorresponds to the physical region in which correlation proper-ties in the motion of the adsorbing particles are modified bythe presence of the fractal interface. While the width of thefractal layer may not coincide with the geometric width of theinterface in the direction of transport, a relationship betweenthese two quantities does exist. Diffusion within the fractal layeris described by means of the constitutive eq 13.

In conclusion, we should just like to mention anotherapproach to model diffusional dynamics within the fractal layerthat is currently under investigation. This approach is basedon the introduction of a multifractal field of diffusivities.39 The

physical explanation of this approach is grounded on theproperties of projected densities of fractal interfaces whichdisplay a multifractal nature.41 The fractional approach discussedin this article and the multifractal description discussed by Gionaand Adrover39 are not antithetical but complementary to eachother. Details of the latter approach are presented elsewhere.39

Acknowledgment. The authors thank R. R. Nigmatullin forsending useful material.

Appendix I. Derivation of Eq 20

The inverse Laplace transform of eq 19 can be expressed interms of fractional derivatives15 by recalling that a basic propertyof the Laplace transform of a -order fractional derivative ( >0) of a continuous function f(t) with vanishing initial conditionsis15

where D represents the fractional derivative of noninteger order, > 0, of the function f(t) with respect to time. By applyingeq 41 to eq 19, one obtains

The explicit expression of the flux, eq 19, is therefore given by

with a noninteger positive order depending on the adsorbentfractal dimension and on the topological dimension, ) df -dT.

The fractional derivative, eq 42, can also be expressed interms of the Riemann-Liouville integral operator. To do this,we recall that the integral form of the fractional derivative

D-qf(t) of a function f(t), given that q > 0, is defined as

where (q) is the gamma function of argument q.In the case of positive , the Laplace transform ofDf(t) can

be expressed as

where q ) m - > 0 and m is the smallest integer greaterthan .

Given that 0 e e 1, by virtue of the definition ) df -dT, it follows that m ) 1, and therefore q ) 1 - .

By applying eqs 43 and 44 to eq 42, it follows that

Equation 45 is the explicit representation in the time domainof the flux-concentration constitutive equation at the interface,

x ) 0.

Appendix II. Laplace Inversion of Eqs 23 and 27

The inverse Laplace transform of eq 23 can be obtainedanalytically by means of the complex inversion formula. To

simplify the notation, let us indicate a ) D F0/K > 0, so thateq 23 becomes

where ) (1 - )/2 ) (1 -D + dT)/2 > 0. If-1 is a multipleof 2, then eq 46 has only one singularity (a branching point) ats ) 0. In the case of the Koch curve considered, ) 1/2, sothat ) 1/4.

In this case, it can easily be seen that the term s + a * 0for every value ofs in the complex plane. In fact, let us assume

that there exists a value s* satisfying the equation s*

+ a ) 0.Then its th power must be a real negative number as a is areal positive number. Let us write s* ) rexp(i). Its power is

s*

) r exp(i) ) r (cos() + i sin()). Since s* is a real

number, sin() ) 0. Therefore, if 1/ is a multiple of 2, itfollows that sin() ) 0. In this case, it follows that /e admitsno singularities other than s ) 0, the square root of a real numberalways being non-negative.

By using the complex inversion formula, the inverse of theLaplace transform of eq 46 can be expressed by the contourintegral in the complex plane

where x0 is a generic real abscissa in the half-plane ofconvergency, Re[s] > 0. The residual of/e at s ) 0 is givenby limsf0 a/[(s + a)] ) 1.

t) - D F *

c

x|x)0 -D F *

c

x|x)0 ) h(Kc - )|x)0 (40)

L [Df(t)] ) sL [f(t)] (41)

L [J]x)0 ) -DF0L [D(cx)x)0]

J|x)0 ) -DF0[D(cx)x)0] (42)

D-q

f(t) )1

(q)0

t(t- )q-1f() d (43)

L [Df(t)] ) L [D

m[D

-qf(t)]] (44)

J|x)0 ) -DF0

(1 - )

t0

t(t- )-(

c

x)x)0 d (45)

(s)

e

)a

s(s + a)(46)

(t)

e

)a

2i

x0-i

x0+iexp(ts)

1

s(s + a)ds



10/10

By using the Cauchy theorem on a contour of the complexplane excluding the singularity s ) 0, Figure 10, and lettingthe radius R of the arc of circumference CR tend to infinity and, the radius of circumference surrounding s ) 0, tend to zero,it follows that

where on the branch CI we can replace s ) u2 exp(i) ) -u2,- < u e 0, and on the branch CII, s ) u2 exp(-i) ) -u2,0 < u e , and on the branch CII, s ) u2 exp(-i) ) -u2, 0< u e . These substitutions lead to the following expression:

which can be expressed, after some algebraic manipulation, inthe form of eq 24.

The inverse Laplace transform of eq 27, can be derived inthe same way. By substituting eq 26 into eq 27, we have

where A ) D F0/K2 and B ) D 0/K2. Since ) 1/4, there isonly one singularity at s ) 0, as discussed above. The residualat s ) 0 is 1. We thus have, as above,

By replacing on CI s ) u2 exp(i) ) -u2, with - < u e 0,and on the CII, s ) u2 exp(-i) ) -u2, with 0 < u e , eq 49becomes

from which eq 28 follows after some algebra.

References and Notes

(1) Seri-Levy, A.; Avnir, D. J. Phys. Chem. 1993, 97, 10380.(2) Delahay, P.; Trachtenberg, I. J. Am. Chem. Soc. 1957, 79, 2355.(3) Delahay, P.; Fike, C. T. J. Am. Chem. Soc. 1958, 80, 2628.(4) Delahay, P.; Mohilner, D. M. J. Am. Chem. Soc. 1962, 84, 4247.(5) Sapoval, B. In Fractals and Disordered Systems; Bunde, A., Havlin,

S., Eds.; Springer Verlag: Berlin, 1991.(6) Larsen, A. E.; Grier, D. G.; Halsey, T. C. Fractals 1994, 2, 191.(7) Pajkossy, T. J. Electroanal. Chem. 1991, 300, 1.(8) Nyikos, L.; Pajkossy, T. Electrochim. Acta 1985, 30, 1533.(9) Bates, J. B.; Chu, Y. T.; Stribling, W. T. Phys. ReV. Lett. 1988,

60, 627.

(10) Kaplan, T.; Gray, L. J.; Liu, S. H. Phys. ReV. B 1987, 35, 5379.(11) Pajkossy, T.; Nyikos, L. Electrochim. Acta 1989, 34, 181.(12) Chassaing, E.; Sapoval, B.; Daccord, G.; Lenormand, R. J.

Electroanal. Chem. 1990, 279, 67.(13) Le Mehaute, A. J. Stat. Phys. 1984, 36, 665.(14) Oldham, K. B.; Spanier, J. The Fractional Calculus; Academic

Press: New York, 1974.(15) Miller, K. S.; Ross, B. An Introduction to the Fractional Calculus

and Fractional Differential Equations; J. Wiley & Sons: New York, 1993.(16) Giona, M.; Roman, H. E. J. Phys. A 1992, 25, 2093.(17) Roman, H. E.; Giona, M. J. Phys. A 1992, 25, 2107.(18) Nigmatullin, R. R. Ph.D. Thesis, Kazan State University, Kazan,

1992 (in Russian).(19) Nigmatullin, R. R. Theor. Math. Phys. 1992, 90, 354 (in Russian).(20) Rabotnov, Yu. N. Elements of Hereditary Solid Mechanics; Mir

Publ.: Moscow, 1980.(21) Nigmatullin, R. R. Phys. Status Solidi 1984, 123, 739.

(22) Nigmatullin, R. R. Phys. Status Solidi 1984, 124, 389.(23) Nigmatullin, R. R. Phys. Status Solidi 1986, 133, 425.(24) Pfeifer, P.; Obert, M.; Cole, M. W. Proc. R. Soc. London A 1989,

423, 169.(25) Fripiat, J. J.; Gatineau, L.; Van Damme, H. Langmuir 1986, 2, 562.(26) Pajkossy, T.; Nyikos, L. Electrochim. Acta 1989, 34, 171.(27) Borosy, A. P.; Nyikos, L.; Pajkossy, T. Electrochim. Acta 1991,

36, 163.(28) Goddard, J. D. Ind. Chem. Res. 1992, 31, 713.(29) Nonnenmacher, T. F. J. Phys. A 1990, 23, L697.(30) Giona, M.; Cerbelli, S.; Roman, H. E. Physica A 1992, 191, 449.(31) Smit, W.; de Vries, H. Rheol. Acta 1970, 9, 525.(32) Bagley, R. L. J. Rheol. 1983, 27, 201.(33) Rogers, L. J. Rheol. 1983, 27, 351.(34) Nonnenmacher, T. F.; Metzler, R. Fractals 1995, 3, 557.(35) Hilfer, R.; Anton, L. Phys. ReV. E 1995, 51, R848.(36) Tatom, F. B. Fractals 1995, 3, 217.

(37) Hilfer, R. Fractals 1995, 3, 549.(38) Giona, M.; Schwalm, W. A.; Adrover, A.; Schwalm, M. K. Chem.

Eng. J., in press.(39) Giona, M.; Adrover, A. First International Conference on Chaos

and Fractals in Chemical Engineering CFIC96, Rome, 2-5 September,1996.

(40) Giona, M.; Giustiniani, A.; Patierno, O. First International Confer-ence on Chaos and Fractals in Chemical Engineering CFIC96, Rome, 2 -5September, 1996.

(41) Gutfraind, R.; Sheintuch, M.; Avnir, D. J. Chem. Phys. 1991, 95,6100.

(42) Crank, J. The Mathematics of Diffusion; Clarendon Press: Oxford,1986.

(43) Ruthven, D. Principles of Adsorption and Adsorption Processes;J. Wiley & Sons: New York, 1984.

JP961518L

(t)

e

) 1 -a

2i

C1exp(ts)

1

s(s + a)ds -

a

2i

CIIexp(ts)

1

s(s + a)ds

(t)

e

) 1 +a

i0 exp(-tu

2)

u[u2

(cos() + i sin()) + a]du -

a

i0 exp(-tu

2)

u[u2(cos() - i sin()) + a]

du (47)

(s)

e

)As + B

s(s + As +B)(48)

(t)

e

) 1 -a

2i

C1exp(ts)

As + B

s(s + As + B)ds -

a2iCII exp(ts) As

+Bs(s + As + B)

ds (49)

(t)

e

) 1 +1

i0exp(-tu

2)

u

Aui + B

(iu + Aui +B)du -

1

i0exp(-tu

2)

u

-Aui +B

(-iu + -Aui + B)du (50)


absorption kinetics on fractal surfaces

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